meta (version 4.9-9)

# metacor: Meta-analysis of correlations

## Description

Calculation of fixed and random effects estimates for meta-analyses with correlations; inverse variance weighting is used for pooling.

## Usage

metacor(
cor,
n,
studlab,
data = NULL,
subset = NULL,
exclude = NULL,
sm = gs("smcor"),
level = gs("level"),
level.comb = gs("level.comb"),
comb.fixed = gs("comb.fixed"),
comb.random = gs("comb.random"),
hakn = gs("hakn"),
method.tau = gs("method.tau"),
method.tau.ci = if (method.tau == "DL") "J" else "QP",
tau.preset = NULL,
TE.tau = NULL,
tau.common = gs("tau.common"),
prediction = gs("prediction"),
level.predict = gs("level.predict"),
null.effect = 0,
method.bias = gs("method.bias"),
backtransf = gs("backtransf"),
title = gs("title"),
complab = gs("complab"),
outclab = "",
byvar,
bylab,
print.byvar = gs("print.byvar"),
byseparator = gs("byseparator"),
keepdata = gs("keepdata"),
control = NULL
)

## Arguments

cor

Correlation.

n

Number of observations.

studlab

An optional vector with study labels.

data

An optional data frame containing the study information, i.e., cor and n.

subset

An optional vector specifying a subset of studies to be used.

exclude

An optional vector specifying studies to exclude from meta-analysis, however, to include in printouts and forest plots.

sm

A character string indicating which summary measure ("ZCOR" or "COR") is to be used for pooling of studies.

level

The level used to calculate confidence intervals for individual studies.

level.comb

The level used to calculate confidence intervals for pooled estimates.

comb.fixed

A logical indicating whether a fixed effect meta-analysis should be conducted.

comb.random

A logical indicating whether a random effects meta-analysis should be conducted.

hakn

A logical indicating whether the method by Hartung and Knapp should be used to adjust test statistics and confidence intervals.

method.tau

A character string indicating which method is used to estimate the between-study variance $$\tau^2$$ and its square root $$\tau$$. Either "DL", "PM", "REML", "ML", "HS", "SJ", "HE", or "EB", can be abbreviated.

method.tau.ci

A character string indicating which method is used to estimate the confidence interval of $$\tau^2$$ and $$\tau$$. Either "QP", "BJ", or "J", or "", can be abbreviated.

tau.preset

Prespecified value for the square root of the between-study variance $$\tau^2$$.

TE.tau

Overall effect used to estimate the between-study variance tau-squared.

tau.common

A logical indicating whether tau-squared should be the same across subgroups.

prediction

A logical indicating whether a prediction interval should be printed.

level.predict

The level used to calculate prediction interval for a new study.

null.effect

A numeric value specifying the effect under the null hypothesis.

method.bias

A character string indicating which test is to be used. Either "rank", "linreg", or "mm", can be abbreviated. See function metabias

backtransf

A logical indicating whether results for Fisher's z transformed correlations (sm = "ZCOR") should be back transformed in printouts and plots. If TRUE (default), results will be presented as correlations; otherwise Fisher's z transformed correlations will be shown.

title

Title of meta-analysis / systematic review.

complab

Comparison label.

outclab

Outcome label.

byvar

An optional vector containing grouping information (must be of same length as event.e).

bylab

A character string with a label for the grouping variable.

print.byvar

A logical indicating whether the name of the grouping variable should be printed in front of the group labels.

byseparator

A character string defining the separator between label and levels of grouping variable.

keepdata

A logical indicating whether original data (set) should be kept in meta object.

control

An optional list to control the iterative process to estimate the between-study variance $$\tau^2$$. This argument is passed on to rma.uni.

## Value

An object of class c("metacor", "meta") with corresponding print, summary, and forest functions. The object is a list containing the following components:

cor, n, studlab, exclude,

As defined above.

sm, level, level.comb,

As defined above.

comb.fixed, comb.random,

As defined above.

hakn, method.tau, method.tau.ci,

As defined above.

tau.preset, TE.tau, method.bias,

As defined above.

method.bias, tau.common, title, complab, outclab,

As defined above.

byvar, bylab, print.byvar, byseparator

As defined above.

TE, seTE

Either Fisher's z transformation of correlations (sm = "ZCOR") or correlations (sm="COR") for individual studies.

lower, upper

Lower and upper confidence interval limits for individual studies.

zval, pval

z-value and p-value for test of effect in individual studies.

w.fixed, w.random

Weight of individual studies (in fixed and random effects model).

TE.fixed, seTE.fixed

Estimated overall effect (Fisher's z transformation of correlation or correlation) and standard error (fixed effect model).

lower.fixed, upper.fixed

Lower and upper confidence interval limits (fixed effect model).

zval.fixed, pval.fixed

z-value and p-value for test of overall effect (fixed effect model).

TE.random, seTE.random

Estimated overall effect (Fisher's z transformation of correlation or correlation) and standard error (random effects model).

lower.random, upper.random

Lower and upper confidence interval limits (random effects model).

zval.random, pval.random

z-value or t-value and corresponding p-value for test of overall effect (random effects model).

prediction, level.predict

As defined above.

seTE.predict

Standard error utilised for prediction interval.

lower.predict, upper.predict

Lower and upper limits of prediction interval.

k

Number of studies combined in meta-analysis.

Q

Heterogeneity statistic Q.

df.Q

Degrees of freedom for heterogeneity statistic.

pval.Q

P-value of heterogeneity test.

tau2

Between-study variance $$\tau^2$$.

se.tau2

Standard error of $$\tau^2$$.

lower.tau2, upper.tau2

Lower and upper limit of confidence interval for $$\tau^2$$.

tau

Square-root of between-study variance $$\tau$$.

lower.tau, upper.tau

Lower and upper limit of confidence interval for $$\tau$$.

H

Heterogeneity statistic H.

lower.H, upper.H

Lower and upper confidence limit for heterogeneity statistic H.

I2

Heterogeneity statistic I$$^2$$.

lower.I2, upper.I2

Lower and upper confidence limit for heterogeneity statistic I$$^2$$.

Rb

Heterogeneity statistic R$$_b$$.

lower.Rb, upper.Rb

Lower and upper confidence limit for heterogeneity statistic R$$_b$$.

df.hakn

Degrees of freedom for test of effect for Hartung-Knapp method (only if hakn = TRUE).

method

Pooling method: "Inverse".

bylevs

Levels of grouping variable - if byvar is not missing.

TE.fixed.w, seTE.fixed.w

Estimated effect and standard error in subgroups (fixed effect model) - if byvar is not missing.

lower.fixed.w, upper.fixed.w

Lower and upper confidence interval limits in subgroups (fixed effect model) - if byvar is not missing.

zval.fixed.w, pval.fixed.w

z-value and p-value for test of effect in subgroups (fixed effect model) - if byvar is not missing.

TE.random.w, seTE.random.w

Estimated effect and standard error in subgroups (random effects model) - if byvar is not missing.

lower.random.w, upper.random.w

Lower and upper confidence interval limits in subgroups (random effects model) - if byvar is not missing.

zval.random.w, pval.random.w

z-value or t-value and corresponding p-value for test of effect in subgroups (random effects model) - if byvar is not missing.

w.fixed.w, w.random.w

Weight of subgroups (in fixed and random effects model) - if byvar is not missing.

df.hakn.w

Degrees of freedom for test of effect for Hartung-Knapp method in subgroups - if byvar is not missing and hakn = TRUE.

n.e.w

Number of observations in experimental group in subgroups - if byvar is not missing.

n.c.w

Number of observations in control group in subgroups - if byvar is not missing.

k.w

Number of studies combined within subgroups - if byvar is not missing.

k.all.w

Number of all studies in subgroups - if byvar is not missing.

Q.w.fixed

Overall within subgroups heterogeneity statistic Q (based on fixed effect model) - if byvar is not missing.

Q.w.random

Overall within subgroups heterogeneity statistic Q (based on random effects model) - if byvar is not missing (only calculated if argument tau.common is TRUE).

df.Q.w

Degrees of freedom for test of overall within subgroups heterogeneity - if byvar is not missing.

pval.Q.w.fixed

P-value of within subgroups heterogeneity statistic Q (based on fixed effect model) - if byvar is not missing.

pval.Q.w.random

P-value of within subgroups heterogeneity statistic Q (based on random effects model) - if byvar is not missing.

Q.b.fixed

Overall between subgroups heterogeneity statistic Q (based on fixed effect model) - if byvar is not missing.

Q.b.random

Overall between subgroups heterogeneity statistic Q (based on random effects model) - if byvar is not missing.

df.Q.b

Degrees of freedom for test of overall between subgroups heterogeneity - if byvar is not missing.

pval.Q.b.fixed

P-value of between subgroups heterogeneity statistic Q (based on fixed effect model) - if byvar is not missing.

pval.Q.b.random

P-value of between subgroups heterogeneity statistic Q (based on random effects model) - if byvar is not missing.

tau.w

Square-root of between-study variance within subgroups - if byvar is not missing.

H.w

Heterogeneity statistic H within subgroups - if byvar is not missing.

lower.H.w, upper.H.w

Lower and upper confidence limit for heterogeneity statistic H within subgroups - if byvar is not missing.

I2.w

Heterogeneity statistic I$$^2$$ within subgroups - if byvar is not missing.

lower.I2.w, upper.I2.w

Lower and upper confidence limit for heterogeneity statistic I$$^2$$ within subgroups - if byvar is not missing.

keepdata

As defined above.

data

Original data (set) used in function call (if keepdata = TRUE).

subset

Information on subset of original data used in meta-analysis (if keepdata = TRUE).

call

Function call.

version

Version of R package meta used to create object.

## Details

Fixed effect and random effects meta-analysis of correlations based either on Fisher's z transformation of correlations (sm = "ZCOR") or direct combination of (untransformed) correlations (sm = "COR") (see Cooper et al., p264-5 and p273-4). Only few statisticians would advocate the use of untransformed correlations unless sample sizes are very large (see Cooper et al., p265). The artificial example given below shows that the smallest study gets the largest weight if correlations are combined directly because the correlation is closest to 1.

Default settings are utilised for several arguments (assignments using gs function). These defaults can be changed for the current R session using the settings.meta function.

Furthermore, R function update.meta can be used to rerun a meta-analysis with different settings.

### Estimation of between-study variance

The following methods to estimate the between-study variance $$\tau^2$$ are available:

• DerSimonian-Laird estimator (method.tau = "DL")

• Paule-Mandel estimator (method.tau = "PM")

• Restricted maximum-likelihood estimator (method.tau = "REML")

• Maximum-likelihood estimator (method.tau = "ML")

• Hunter-Schmidt estimator (method.tau = "HS")

• Sidik-Jonkman estimator (method.tau = "SJ")

• Hedges estimator (method.tau = "HE")

• Empirical Bayes estimator (method.tau = "EB")

See metagen for more information on these estimators.

### Confidence interval for the between-study variance

The following methods to calculate a confidence interval for $$\tau^2$$ and $$\tau$$ are available.

 Argument Method method.tau.ci = "J" Method by Jackson method.tau.ci = "BJ" Method by Biggerstaff and Jackson
See metagen for more information on these methods. No confidence intervals for $$\tau^2$$ and $$\tau$$ are calculated if method.tau.ci = "".

### Hartung-Knapp method

Hartung and Knapp (2001) and Knapp and Hartung (2003) proposed an alternative method for random effects meta-analysis based on a refined variance estimator for the treatment estimate. Simulation studies (Hartung and Knapp, 2001; IntHout et al., 2014; Langan et al., 2019) show improved coverage probabilities compared to the classic random effects method. However, in rare settings with very homogeneous treatment estimates, the Hartung-Knapp method can be anti-conservative (Wiksten et al., 2016). The Hartung-Knapp method is used if argument hakn = TRUE.

### Prediction interval

A prediction interval for the proportion in a new study (Higgins et al., 2009) is calculated if arguments prediction and comb.random are TRUE. Note, the definition of prediction intervals varies in the literature. This function implements equation (12) of Higgins et al., (2009) which proposed a t distribution with K-2 degrees of freedom where K corresponds to the number of studies in the meta-analysis.

### Subgroup analysis

Argument byvar can be used to conduct subgroup analysis for a categorical covariate. The metareg function can be used instead for more than one categorical covariate or continuous covariates.

### Presentation of meta-analysis results

Internally, both fixed effect and random effects models are calculated regardless of values choosen for arguments comb.fixed and comb.random. Accordingly, the estimate for the random effects model can be extracted from component TE.random of an object of class "meta" even if argument comb.random = FALSE. However, all functions in R package meta will adequately consider the values for comb.fixed and comb.random. E.g. functions print.meta and forest.meta will not print results for the random effects model if comb.random = FALSE.

## References

Cooper H, Hedges LV, Valentine JC (2009): The Handbook of Research Synthesis and Meta-Analysis, 2nd Edition. New York: Russell Sage Foundation

DerSimonian R & Laird N (1986): Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177--88

Hartung J & Knapp G (2001): On tests of the overall treatment effect in meta-analysis with normally distributed responses. Statistics in Medicine, 20, 1771--82

Higgins JPT, Thompson SG, Spiegelhalter DJ (2009): A re-evaluation of random-effects meta-analysis. Journal of the Royal Statistical Society: Series A, 172, 137--59

IntHout J, Ioannidis JPA, Borm GF (2014): The Hartung-Knapp-Sidik-Jonkman method for random effects meta-analysis is straightforward and considerably outperforms the standard DerSimonian-Laird method. BMC Medical Research Methodology, 14, 25

Knapp G & Hartung J (2003): Improved tests for a random effects meta-regression with a single covariate. Statistics in Medicine, 22, 2693--710

Langan D, Higgins JPT, Jackson D, Bowden J, Veroniki AA, Kontopantelis E, et al. (2019): A comparison of heterogeneity variance estimators in simulated random-effects meta-analyses. Research Synthesis Methods, 10, 83--98

Viechtbauer W (2010): Conducting Meta-Analyses in R with the Metafor Package. Journal of Statistical Software, 36, 1--48

Wiksten A, R<U+00FC>cker G, Schwarzer G (2016): Hartung-Knapp method is not always conservative compared with fixed-effect meta-analysis. Statistics in Medicine, 35, 2503--15

update.meta, metacont, metagen, print.meta

## Examples

# NOT RUN {
m1 <- metacor(c(0.85, 0.7, 0.95), c(20, 40, 10))

# Print correlations (back transformed from Fisher's z
# transformation)
#
m1

# Print Fisher's z transformed correlations
#
print(m1, backtransf = FALSE)

# Forest plot with back transformed correlations
#
forest(m1)

# Forest plot with Fisher's z transformed correlations
#
forest(m1, backtransf = FALSE)

m2 <- update(m1, sm = "cor")
m2

# Identical forest plots (as back transformation is the identity
# transformation)
# forest(m2)
# forest(m2, backtransf = FALSE)

# }